Interferometric iterative technique with bandwidth and numerical-aperture dependency

Abstract

An interferometric intensity equation includes parameters that depend on bandwidth and numerical aperture. An error function based on the difference between actual intensities produced by interferometry and the intensities predicted by the equation is minimized iteratively with respect to the parameters. The scan positions (i.e., the step sizes between frames) that minimized the error function are then used to calculate the phase for each pixel, from which the height can also be calculated in conventional manner. As a result, the phase map generated by the procedure is corrected to a degree of precision significantly better than previously possible.

1. A method of reducing errors in measurements performed with an interference microscope, the method comprising the following steps:
 employing an equation expressing intensity as a function of planar coordinates, optical path difference, phase, and a set of equation parameters reflecting an implicit dependence on illumination spectral bandwidth and numerical aperture, said equation parameters having an explicit dependence on planar coordinates, optical path difference and phase; 
 replacing said equation parameters with new phase-dependent parameters such that said explicit dependence of the equation parameters on planar coordinates is decoupled from the dependence on optical path difference; 
 fitting said equation to global interference data acquired by varying said optical path difference, thereby producing an optimal set of said equation parameters, said fitting step being an iterative process wherein said equation parameters are estimated to a predetermined iterative threshold of convergence; and 
 calculating a modified phase value using said optimal set of equation parameters. 
 
     
     
       2. The method of  claim 1 , further including the step of calculating a height map of a sample based on said modified phase. 
     
     
       3. The method of  claim 1 , wherein the calculating step attenuates an impact of noise in the measurements due to mechanical vibrations present at the interference microscope. 
     
     
       4. The method of  claim 2 , wherein the calculating step attenuates an impact of noise in the height map due to mechanical vibrations present at the interference microscope. 
     
     
       5. The method of  claim 1 , wherein said interference data acquired by varying the optical path difference are taken symmetrically around a best focus position. 
     
     
       6. The method of  claim 1 , wherein said interference data acquired by varying the optical path difference are produced using a light source including a bandpass filter of approximately 20-nm bandwidth. 
     
     
       7. The method of  claim 1  wherein said interference data are acquired over three to twelve frames separated by a nominal phase step of 
       
         
           
             
               
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       8. The method of  claim 1 , wherein said interference microscope is operated so as to produce about five fringes within a field of view of an objective of the microscope. 
     
     
       9. The method of  claim 1 , wherein said step of varying the optical path difference is carried out with a vertical scan. 
     
     
       10. The method of  claim 1 , wherein the method is used to measure a magnetic read-write head for data storage. 
     
     
       11. The method of  claim 1 , wherein the method is used to measure an electronic substrate in a semiconductor component. 
     
     
       12. The method of  claim 1 , wherein the method is used to measure a plastic electronics component. 
     
     
       13. The method of  claim 1 , wherein the method is used to measure a solar energy conversion device. 
     
     
       14. The method of  claim 1 , wherein the method is used to measure a micro-mirror array. 
     
     
       15. The method of  claim 1 , wherein the method is used to measure a MEMS device. 
     
     
       16. An interferometric iterative procedure comprising:
 selecting an equation expressing intensity as a function of planar coordinates, optical path difference, phase, and a set of bandwidth-dependent and numerical-aperture-dependent equation parameters, said equation parameters having an explicit dependence on planar coordinates, optical path difference and phase; 
 replacing said equation parameters with new phase-dependent parameters such that said explicit dependence of the equation parameters on planar coordinates is decoupled from the dependence on optical path difference; 
 fitting said equation to global interference data acquired by varying said optical path difference, thereby producing an optimal set of said equation parameters; and 
 using said optimal set of equation parameters to calculate a corrected phase. 
 
     
     
       17. The method of  claim 16 , wherein said fitting step includes establishing an error function based on said equation and minimizing the error function with respect to said set of equation parameters over the global interference data, thereby producing the optimal set of equation parameters. 
     
     
       18. The method of  claim 17 , wherein the step of minimizing the error function involves:
 (a) performing a first sub-step of minimizing the error function with respect to a first subset of said new phase-dependent parameters; 
 (b) producing an updated error function with a first sub-set of the new-phase-dependent parameters that minimized the error function during said first sub-step; 
 (c) performing a second sub-step of minimizing the updated error function with respect to a second subset of said new phase-dependent parameters; 
 (d) producing a further updated error function with a second sub-set of the new phase-dependent parameters that minimized the error function during said second sub-step; and 
 (e) iteratively repeating steps (a) through (d) until a predetermined iterative condition is met. 
 
     
     
       19. The method of  claim 18 , further including the step of calculating a height map of a sample based on said corrected phase. 
     
     
       20. The method of  claim 17 , wherein said error function is a least-squares representation of a difference between actual intensities produced by an interferometric system and theoretical intensities predicted by said equation expressing intensity as a function of planar coordinates, optical path difference, phase, and said set of bandwidth-dependent and numerical-aperture-dependent equation parameters. 
     
     
       21. The method of  claim 16 , further including the step of calculating a height map of a sample based on said corrected phase. 
     
     
       22. The method of  claim 16 , wherein said interference data acquired by varying the optical path difference are taken symmetrically around a best focus position. 
     
     
       23. The method of  claim 16 , wherein said interference data acquired by varying the optical path difference are produced using a light source including a bandpass filter of approximately 20-nm bandwidth. 
     
     
       24. The method of  claim 16 , wherein said interference data are acquired over three to twelve frames separated by a nominal phase step of 
       
         
           
             
               
                 π 
                 2 
               
               . 
             
           
         
       
     
     
       25. The method of  claim 16 , wherein said interference data are acquired in a system producing about five fringes within a field of view of an objective of the system. 
     
     
       26. The method of  claim 16 , wherein said step of varying the optical path difference is carried out with a vertical scan. 
     
     
       27. The method of  claim 16 , wherein said interference data acquired by varying the optical path difference are taken symmetrically around a best focus position using a light source of approximately 20-nm bandwidth, and the interference data are acquired over three to twelve frames separated by a nominal phase step of 
       
         
           
             
               
                 π 
                 2 
               
               . 
             
           
         
       
     
     
       28. An interferometric method of profiling a sample surface comprising the following steps:
 performing an interferometric measurement of the surface, thereby producing a corresponding set of interferometric data; 
 establishing an error function based on an equation expressing intensity as a function of planar coordinates, optical path difference, phase, and a set of bandwidth-dependent equation parameters, said equation parameters having an explicit dependence on planar coordinates, optical path difference and phase; 
 replacing said equation parameters with new phase-dependent parameters such that said explicit dependence of the equation parameters on planar coordinates is decoupled from the dependence on optical path difference; 
 minimizing said error function with respect to said equation parameters over the set of interferometric data, thereby producing an optimal set of equation parameters; and 
 using said optimal set of equation parameters to calculate a corrected phase. 
 
     
     
       29. The method of  claim 28 , wherein said error function is a least-squares representation of a difference between actual intensities produced by said interferometric measurement and theoretical intensities predicted by said equation expressing intensity as a function of planar coordinates, optical path difference, phase, and said set of bandwidth-dependent equation parameters. 
     
     
       30. The method of  claim 28 , further including the step of calculating a height map of the sample based on said corrected phase. 
     
     
       31. The method of  claim 28 , wherein:
 said error function is a least-squares representation of a difference between actual intensities produced by said interferometric measurement and theoretical intensities predicted by said equation; and 
 said interferometric measurement is carried out by varying said optical path difference with a vertical scan with data-acquisition frames separated by a nominal phase step of 
 
       
         
           
             
               
                 π 
                 2 
               
               . 
             
           
         
       
     
     
       32. An interferometric method of profiling a sample surface comprising the following steps:
 performing an interferometric measurement of the surface, thereby producing a corresponding set of interferometric data; 
 establishing an error function based on an equation expressing intensity as a function of planar coordinates, optical path difference, phase, and a set of numerical-aperture-dependent equation parameters, said equation parameters having an explicit dependence on planar coordinates, optical path difference and phase; 
 replacing said equation parameters with new phase-dependent parameters such that said explicit dependence of the equation parameters on planar coordinates is decoupled from the dependence on optical path difference; 
 minimizing said error function with respect to said equation parameters over said set of interferometric data, thereby producing an optimal set of equation parameters; and 
 using said optimal set of equation parameters to calculate a corrected phase. 
 
     
     
       33. The method of  claim 32 , wherein said error function is a least-squares representation of a difference between actual intensities produced by said interferometric measurement and theoretical intensities predicted by said equation expressing intensity as a function of and said numerical-aperture-dependent parameter, planar coordinates, optical path difference, phase, and said set of numerical-aperture-dependent equation parameters. 
     
     
       34. The method of  claim 32 , further including the step of calculating a height map of the sample based on said corrected phase. 
     
     
       35. The method of  claim 32 , wherein:
 said error function is a least-squares representation of a difference between actual intensities produced by said interferometric measurement and theoretical intensities predicted by said equation; and 
 said interferometric measurement is carried out by varying said optical path difference with a vertical scan with data-acquisition frames separated by a nominal phase step of 
 
       
         
           
             
               
                 π 
                 2 
               
               . 
             
           
         
       
     
     
       36. A method of correcting effects of modulation variations caused by illumination bandwidth and numerical aperture in an interferometric system, the method comprising the following steps:
 developing an equation expressing intensity as a function of planar coordinates, scanner position, phase, and a set of bandwidth-dependent and numerical-aperture-dependent equation parameters, said equation parameters having an explicit dependence on planar coordinates, scanner position and phase; 
 replacing said equation parameters with new phase-dependent parameters such that said explicit dependence of the equation parameters on planar coordinates is decoupled from the dependence on scanner position; 
 establishing an error function as a least-squares representation of a difference between actual intensities produced by said interferometric system and theoretical intensities predicted by said equation; 
 minimizing said error function with respect to said set of equation parameters over global interference data acquired through an interferometric scan, said minimizing step including:
 (a) performing a first sub-step of minimizing the error function with respect to a first subset of said new phase-dependent parameters; 
 (b) producing an updated error function with a first sub-set of the new-phase-dependent parameters that minimized the error function during said first sub-step; 
 (c) performing a second sub-step of minimizing the updated error function with respect to a second subset of said new phase-dependent parameters; 
 (d) producing a further updated error function with a second sub-set of the new phase-dependent parameters that minimized the error function during said second sub-step; and 
 (e) iteratively repeating steps (a) through (d) until a predetermined iterative condition is met; 
 
 thereby producing an optimal set of said equation parameters; 
 using said optimal set of equation parameters to calculate a corrected phase; and 
 calculating a height map of a sample based on said corrected phase.